The homotopy Lie algebra of symplectomorphism groups of 3-fold blow-ups of $(S^2 \times S^2, σ_{std} \oplus σ_{std}) $ (1702.03572v2)
Abstract: We consider the 3-point blow-up of the manifold $ (S2 \times S2, \sigma \oplus \sigma)$ where $\sigma$ is the standard symplectic form which gives area 1 to the sphere $S2$, and study its group of symplectomorphisms $\rm{Symp} ( S2 \times S2 #\, 3\overline{\mathbb C\mathbb P}\,!2, \omega)$. So far, the monotone case was studied by J. Evans and he proved that this group is contractible. Moreover, J. Li, T. J. Li and W. Wu showed that the group Symp$_{h}(S2 \times S2 #\, 3\overline{ \mathbb C\mathbb P}\,!2,\omega) $ of symplectomorphisms that act trivially on homology is always connected and recently they also computed its fundamental group. We describe, in full detail, the rational homotopy Lie algebra of this group. We show that some particular circle actions contained in the symplectomorphism group generate its full topology. More precisely, they give the generators of the homotopy graded Lie algebra of $\rm{Symp} (S2 \times S2 #\, 3\overline{ \mathbb C\mathbb P}\,!2, \omega)$. Our study depends on Karshon's classification of Hamiltonian circle actions and the inflation technique introduced by Lalonde-McDuff. As an application, we deduce the rank of the homotopy groups of $\rm{Symp}({\mathbb C\mathbb P}2 #\, 5\overline{\mathbb C\mathbb P}\,!2, \tilde \omega)$, in the case of small blow-ups.